measuring cycle time through the use of the ... - Semantic Scholar
Proceedings of the 2014 Winter Simulation Conference A. Tolk, S. Y. Diallo, I. O. Ryzhov, L. Yilmaz, S. Buckley, and J. A. Miller, eds.
MEASURING CYCLE TIME THROUGH THE USE OF THE QUEUING THEORY FORMULA (G/G/M) DJ Kim Lixin Wang Robert Havey Industrial Engineering Department Micron Technology, Inc. 9600 Godwin Drive Manassas, VA 20110, USA ABSTRACT Semiconductor manufacturers are required to reduce their product cycle times since many product embedded semiconductor devices often have a very short life cycle. One way to reduce cycle time is to purchase extra manufacturing tools. However, these tools cost several millions of dollars and facility space is limited. Another way to reduce cycle time is to improve performance of the critical tools. The second option is less costly and provides a significant cost savings for manufactures, which leads them to maximize efficiency. In order to determine which tools are critical and require analytical resources to optimize their performance, a system is needed to prioritize which are the critical tools. This paper will focus on Kendall’s Classification of Queues, and it will focus on the G/G/m Queue (general distribution arrival process / general distribution service process / m servers) (Kendall 1953). 1
QUEUING FORMULAS
Queuing theory is the mathematical study of waiting lines or queues. A facility can be conceptualized as a set of products (wafers) traveling through a network of queues whose servers are tools. Optimizing the variation of arriving work in progress (WIP), WIP processing times, tool repair time, and the number of qualified tools will improve cycle time for the system and increase throughput of the critical tools. Kendall’s classification of a queuing station (A/B/m) (Kendall 1953), where: A: Arrival process B: Service process m: number of machines and distributions: M: Exponential (Markovian) distribution G: Completely general distribution D: Constant (Deterministic) distribution M/M/m M/G/m M/D/m G/M/m
978-1-4799-7486-3/14/$31.00 ©2014 IEEE
2414
Kim, Wang, and Havey G/G/m G/D/m D/M/m D/G/m D/D/m
Figure 1: Characterization of a queuing station. The station in Figure 1 can be described using the following parameters: ra: Rate of arrivals in job per unit time ta: Average time between arrivals (ta = 1/ra) ca: Coefficient of variation of inter-arrival times m: Number of machines re: Rate of the station in jobs per unit time ce: Coefficient of variation of effective process times u: Utilization of station (ra/re) and the following measures: CTq: Expected waiting time spent in queue CT: Expected time spent at the process center (queue time plus process time WIPq: Expected WIP (in jobs) in queue WIP: Average WIP level (in jobs) at the station with the following relationships: CT = CTq + te WIP = ra * CT WIPq = ra * CTq If CTq is known, WIP, WIPq and CT can be calculated (Kendall 1953). The Kingman’s equation (Kingman 1961) modified for m servers (Hopp and Spearman 2001, Medhi 1991): 2
2415
1
Kim, Wang, and Havey 1.1
The G/G/m Queue
The G/G/m queue is a completely general distribution of arrival and process times with m servers. No exact performance measures can be written, so approximation is used. Cases where approximation works poorly are where ca and ce are much larger than 1, and u is larger than 0.95 or smaller than 0.1. In addition, the assumptions of the G/G/m queue are First-Come, First-Served, infinite calling population and unlimited queue lengths are allowed. 1.1.1 Notations A: Effective availability b: Weighted average number of lots processed c02: Squared coefficient of variation of natual process time ca2: Squared coefficient of variation of arrivals of lots or batches ce2: Squared coefficient of variation of process time cr2: Squared coefficient of variation of repair time m: Number of qualified machines mr: Average length of mean time to repair (MTTR) t0: Average natural process time te: Mean effective process time u: Average utilization of machines Step Cycle Time Formula (Hopp and Spearman 2001):
2
1
1
1.2
1
Limitations of Cycle Time Formula
The Cycle Time Formula is a static model because the model does not run over time. The accuracy of expected moves is very important for an accurate prediction due to the reentrance flow effect seen in semiconductor manufacturing. The formula also assumes that Part-Step (PS) jobs waiting at queue can be processed by any of the servers (M) (Hopp and Spearman 2001), Figure 2, which may not be correct if all servers are not qualified to process the WIP.
Figure 2: G/G/m job waiting queue.
2416
Kim, Wang, and Havey 2
VALIDATIONS OF INPUT PARAMETERS
Accuracy of cycle time estimation using G/G/m queue heavily relies on the input parameters. One way is to aggregate all variations (lot variation, equipment variation, and processe time variation) into one single parameter Variability (Schelasin 2013b). A backward calculation method is used with formula
Where , U, and are all historical data. This paper adopts a different approach to determine the input parameters. All input parameters without aggregation used in G/G/m queue are generated based on historical data with time span of 1~3 months, namely A, ca, co, cr, m, te, and b. Certain filtering criteria is required to remove outlier data. When upstream tool has long term down, during that time period, time between lot arrivals will be peak points as shown in Figure 3. Thus ca value will go up significantly. Another example is MTTR value. Due to system setup, certain tools log have large number of down event 10%. Since the project was started, the process areas are using the cycle time formula to help them identify the area bottleneck and near bottleneck, and work on methods to improve those workstations. Silo area project work is held to a minimum due to an alignment of process areas that naturally work together based on the process flow. This synergy is helping reduce variability in the line through better communication and a stronger understanding of the impact that line variation has on cycle time. By understanding the components that go into cycle time, if becomes easier to understand how our previous actions have impacted cycle time. The process of understanding the complexities and interrelatedness of the various components that makeup cycle time is allowing for changes in behavior that have negatively impacted cycle time in the past. It also drove home the point that regular feedback was necessary to ensure cycle time does not get out of control in the future. The Cycle Time operating curves show how throughput, utilization and cycle time are connected, but WIP levels still needs to be calculated from these values. Intel developed a method based on the operating curve and Little’s Law (Hopp and Spearman 2001) to sow this relationship called O_L Graph (Li et al. 2005). The methods and formulas described in the Intel paper are being explored to determine if and how they can be used to help improve cycle time without decreasing facility output. REFERENCES Hopp, W. and Spearman, M. L. 2001. Factory Physics. Boston: Irwin. Kendall, D. G. 1953. “Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Methods of the Imbedded Markov Chain.” The Annals of Mathematical Statistics 24 (3): 338-354. Kingman, J. F. C. 1961. “The Single-Server Queue in Heavy Traffic.” Mathematical Proceedings of the Cambridge Philosophical Society 57(4): 902-904. Li, N., L. Zhang, M. Zhang, and L. Zheng. 2005. “Applied Factory Physics Study on Semiconductor Assembly and Test Manufacturing”. In Proceedings of the 2005 IEEE International Symposium on Semiconductor Manufacturing, 307–311. Medhi, J. 1991. Stochastic Models in Queuing Theory. Boston, MA: Academic Press. Schelasin, R. E. A. 2013a. “Capacity Management Using Static Modeling, Queuing Theory, and Performance Curves”. IIE Annual Conference & Expo.
2420
Kim, Wang, and Havey Schelasin, R. E. A. 2013b. “Estimating Wafer Processing Cycle Time Using an Improved G/G/M Queue”. In Proceedings of the 2013 Winter Simulation Conference, edited by R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl, 3789–3795. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc. AUTHOR BIOGRAPHIES DJ KIM is a senior member technical staff in the Industrial Engineering Department at Micron Technology Inc., Virginia. He received M.S. in Engineering Management from Queensland University of Technology, Australia, an MBA from Georgetown University, and has performed 16 years of simulation and modeling experiences in semiconductor industry. He is a member of INFORMS. His e-mail is [email protected]. LIXIN WANG is a senior industrial engineer in the Industrial Engineering department at Micron Technology Inc., Virginia. He received B.S. in Mechanical Engineering from Tsinghua University, Beijing, China and Ph.D. in Industrial and Systems Engineering from Virginia Tech. His interest is mathematical modeling and simulation of semiconductor manufacturing systems. His e-mail is [email protected]. ROBERT HAVEY is an industrial engineer in the Industrial Engineering Department at Micron Technology, Inc. and has worked 12 years in semiconductor and automated material handling systems planning. He received his B.S. in Industrial Engineering from Texas A&M University. His e-mail is [email protected].
2421
MEASURING CYCLE TIME THROUGH THE USE OF THE QUEUING THEORY FORMULA (G/G/M) DJ Kim Lixin Wang Robert Havey Industrial Engineering Department Micron Technology, Inc. 9600 Godwin Drive Manassas, VA 20110, USA ABSTRACT Semiconductor manufacturers are required to reduce their product cycle times since many product embedded semiconductor devices often have a very short life cycle. One way to reduce cycle time is to purchase extra manufacturing tools. However, these tools cost several millions of dollars and facility space is limited. Another way to reduce cycle time is to improve performance of the critical tools. The second option is less costly and provides a significant cost savings for manufactures, which leads them to maximize efficiency. In order to determine which tools are critical and require analytical resources to optimize their performance, a system is needed to prioritize which are the critical tools. This paper will focus on Kendall’s Classification of Queues, and it will focus on the G/G/m Queue (general distribution arrival process / general distribution service process / m servers) (Kendall 1953). 1
QUEUING FORMULAS
Queuing theory is the mathematical study of waiting lines or queues. A facility can be conceptualized as a set of products (wafers) traveling through a network of queues whose servers are tools. Optimizing the variation of arriving work in progress (WIP), WIP processing times, tool repair time, and the number of qualified tools will improve cycle time for the system and increase throughput of the critical tools. Kendall’s classification of a queuing station (A/B/m) (Kendall 1953), where: A: Arrival process B: Service process m: number of machines and distributions: M: Exponential (Markovian) distribution G: Completely general distribution D: Constant (Deterministic) distribution M/M/m M/G/m M/D/m G/M/m
978-1-4799-7486-3/14/$31.00 ©2014 IEEE
2414
Kim, Wang, and Havey G/G/m G/D/m D/M/m D/G/m D/D/m
Figure 1: Characterization of a queuing station. The station in Figure 1 can be described using the following parameters: ra: Rate of arrivals in job per unit time ta: Average time between arrivals (ta = 1/ra) ca: Coefficient of variation of inter-arrival times m: Number of machines re: Rate of the station in jobs per unit time ce: Coefficient of variation of effective process times u: Utilization of station (ra/re) and the following measures: CTq: Expected waiting time spent in queue CT: Expected time spent at the process center (queue time plus process time WIPq: Expected WIP (in jobs) in queue WIP: Average WIP level (in jobs) at the station with the following relationships: CT = CTq + te WIP = ra * CT WIPq = ra * CTq If CTq is known, WIP, WIPq and CT can be calculated (Kendall 1953). The Kingman’s equation (Kingman 1961) modified for m servers (Hopp and Spearman 2001, Medhi 1991): 2
2415
1
Kim, Wang, and Havey 1.1
The G/G/m Queue
The G/G/m queue is a completely general distribution of arrival and process times with m servers. No exact performance measures can be written, so approximation is used. Cases where approximation works poorly are where ca and ce are much larger than 1, and u is larger than 0.95 or smaller than 0.1. In addition, the assumptions of the G/G/m queue are First-Come, First-Served, infinite calling population and unlimited queue lengths are allowed. 1.1.1 Notations A: Effective availability b: Weighted average number of lots processed c02: Squared coefficient of variation of natual process time ca2: Squared coefficient of variation of arrivals of lots or batches ce2: Squared coefficient of variation of process time cr2: Squared coefficient of variation of repair time m: Number of qualified machines mr: Average length of mean time to repair (MTTR) t0: Average natural process time te: Mean effective process time u: Average utilization of machines Step Cycle Time Formula (Hopp and Spearman 2001):
2
1
1
1.2
1
Limitations of Cycle Time Formula
The Cycle Time Formula is a static model because the model does not run over time. The accuracy of expected moves is very important for an accurate prediction due to the reentrance flow effect seen in semiconductor manufacturing. The formula also assumes that Part-Step (PS) jobs waiting at queue can be processed by any of the servers (M) (Hopp and Spearman 2001), Figure 2, which may not be correct if all servers are not qualified to process the WIP.
Figure 2: G/G/m job waiting queue.
2416
Kim, Wang, and Havey 2
VALIDATIONS OF INPUT PARAMETERS
Accuracy of cycle time estimation using G/G/m queue heavily relies on the input parameters. One way is to aggregate all variations (lot variation, equipment variation, and processe time variation) into one single parameter Variability (Schelasin 2013b). A backward calculation method is used with formula
Where , U, and are all historical data. This paper adopts a different approach to determine the input parameters. All input parameters without aggregation used in G/G/m queue are generated based on historical data with time span of 1~3 months, namely A, ca, co, cr, m, te, and b. Certain filtering criteria is required to remove outlier data. When upstream tool has long term down, during that time period, time between lot arrivals will be peak points as shown in Figure 3. Thus ca value will go up significantly. Another example is MTTR value. Due to system setup, certain tools log have large number of down event 10%. Since the project was started, the process areas are using the cycle time formula to help them identify the area bottleneck and near bottleneck, and work on methods to improve those workstations. Silo area project work is held to a minimum due to an alignment of process areas that naturally work together based on the process flow. This synergy is helping reduce variability in the line through better communication and a stronger understanding of the impact that line variation has on cycle time. By understanding the components that go into cycle time, if becomes easier to understand how our previous actions have impacted cycle time. The process of understanding the complexities and interrelatedness of the various components that makeup cycle time is allowing for changes in behavior that have negatively impacted cycle time in the past. It also drove home the point that regular feedback was necessary to ensure cycle time does not get out of control in the future. The Cycle Time operating curves show how throughput, utilization and cycle time are connected, but WIP levels still needs to be calculated from these values. Intel developed a method based on the operating curve and Little’s Law (Hopp and Spearman 2001) to sow this relationship called O_L Graph (Li et al. 2005). The methods and formulas described in the Intel paper are being explored to determine if and how they can be used to help improve cycle time without decreasing facility output. REFERENCES Hopp, W. and Spearman, M. L. 2001. Factory Physics. Boston: Irwin. Kendall, D. G. 1953. “Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Methods of the Imbedded Markov Chain.” The Annals of Mathematical Statistics 24 (3): 338-354. Kingman, J. F. C. 1961. “The Single-Server Queue in Heavy Traffic.” Mathematical Proceedings of the Cambridge Philosophical Society 57(4): 902-904. Li, N., L. Zhang, M. Zhang, and L. Zheng. 2005. “Applied Factory Physics Study on Semiconductor Assembly and Test Manufacturing”. In Proceedings of the 2005 IEEE International Symposium on Semiconductor Manufacturing, 307–311. Medhi, J. 1991. Stochastic Models in Queuing Theory. Boston, MA: Academic Press. Schelasin, R. E. A. 2013a. “Capacity Management Using Static Modeling, Queuing Theory, and Performance Curves”. IIE Annual Conference & Expo.
2420
Kim, Wang, and Havey Schelasin, R. E. A. 2013b. “Estimating Wafer Processing Cycle Time Using an Improved G/G/M Queue”. In Proceedings of the 2013 Winter Simulation Conference, edited by R. Pasupathy, S.-H. Kim, A. Tolk, R. Hill, and M. E. Kuhl, 3789–3795. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc. AUTHOR BIOGRAPHIES DJ KIM is a senior member technical staff in the Industrial Engineering Department at Micron Technology Inc., Virginia. He received M.S. in Engineering Management from Queensland University of Technology, Australia, an MBA from Georgetown University, and has performed 16 years of simulation and modeling experiences in semiconductor industry. He is a member of INFORMS. His e-mail is [email protected]. LIXIN WANG is a senior industrial engineer in the Industrial Engineering department at Micron Technology Inc., Virginia. He received B.S. in Mechanical Engineering from Tsinghua University, Beijing, China and Ph.D. in Industrial and Systems Engineering from Virginia Tech. His interest is mathematical modeling and simulation of semiconductor manufacturing systems. His e-mail is [email protected]. ROBERT HAVEY is an industrial engineer in the Industrial Engineering Department at Micron Technology, Inc. and has worked 12 years in semiconductor and automated material handling systems planning. He received his B.S. in Industrial Engineering from Texas A&M University. His e-mail is [email protected].
2421
